For each fixed k≥6 the maximum number h_k(n) of k-holes in n-point sets with no (k+1)-hole satisfies (c1/k)^⌊k/3⌋ n^⌊k/3⌋ ≤ h_k(n) ≤ (c2/k)^⌈k/2⌉ n^⌈k/2⌉ for absolute constants c1,c2>0.
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On the maximum number of $k$-holes in point sets with no $(k + 1)$-hole
For each fixed k≥6 the maximum number h_k(n) of k-holes in n-point sets with no (k+1)-hole satisfies (c1/k)^⌊k/3⌋ n^⌊k/3⌋ ≤ h_k(n) ≤ (c2/k)^⌈k/2⌉ n^⌈k/2⌉ for absolute constants c1,c2>0.